"Additivity" versus "Maxitivity" at the heart of the paradoxical and efficient nature of Statistics

Abstract

Unlike the Probability Theory based on additivity, Statistical Inference seems to hesitate between "Additivity" and a so-called "Maxitivity" approach. After a brief overview of three types of principles for any (parametric) statistical theory and the proof that these principles are mutually exclusive, the paper shows that two kinds of support measures are conceivable, an additive one and a maxitive one (based on maximization operators). Unfortunately, none of them is able to cope with the ignorance part of the statistical experiment and, in the meantime, with the partial information given through the structure of the data. To conclude, the author promotes the combined use of both approaches, as an efficient middle-of-the-road position for the statistician.